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| diagram of root structure: Anatomy of Flowering Plants Paula J. Rudall, 2007-03-15 In the 2007 third edition of her successful textbook, Paula Rudall provides a comprehensive yet succinct introduction to the anatomy of flowering plants. Thoroughly revised and updated throughout, the book covers all aspects of comparative plant structure and development, arranged in a series of chapters on the stem, root, leaf, flower, seed and fruit. Internal structures are described using magnification aids from the simple hand-lens to the electron microscope. Numerous references to recent topical literature are included, and new illustrations reflect a wide range of flowering plant species. The phylogenetic context of plant names has also been updated as a result of improved understanding of the relationships among flowering plants. This clearly written text is ideal for students studying a wide range of courses in botany and plant science, and is also an excellent resource for professional and amateur horticulturists. |
| diagram of root structure: Principles of Modern Biology Douglas Marsland, 1945 The cell; Multicellular plants; Multicellular animals, especially man; Heredity and evolution. |
| diagram of root structure: Tree Roots in the Built Environment John Roberts, Nick Jackson, Mark Smith, Centre for Ecology and Hydrology (Great Britain), 2006-06-14 This publication sets out a comprehensive review of tree root biology and covers a broad range of practical issues that need to be considered in order to grow trees successfully in our towns and cities and to realise the significant benefits they provide in built environments. Topics covered include: soil condition and roots; improving tree root growth in urban soils; water supply and drought amelioration for amenity trees; coping with soil contamination; protecting trees during excavation and good trenching practice; control of damage to tree roots on construction sites; tree root damage to buildings and pavements, sewers, drains and pipes; research needs and sustainability issues. |
| diagram of root structure: Inanimate Life George M. Briggs, 2021-07-16 |
| diagram of root structure: Methods of Studying Root Systems W. Böhm, 2012-12-06 Root research under natural field conditions is still a step-child of science. The reason for this is primarily methodological. The known methods are tedious, time consuming, and the accuracy of their results is often not very great. Many research workers have been discouraged by doing such root studies. The need for more information on the development and distribution of plant roots in different soils under various ecological conditions is, however, obvious in many ecological disciplines. Especially the applied botanical sciences such as agriculture, horticulture, and forestry are interested in obtaining more data on plant roots in the soil. This book will give a survey of existing methods in ecological root research. Primarily field methods are presented; techniques for pot experiments are described only so far as they are important for solving ecological problems. Laboratory methods for studying root physiology are not covered in this book. Scientific publications on roots are scattered in many different journals published all over the world. By working through the international root literature I found that about ten thousand papers on root ecology have been published at the present. This is not very much compared with the immense literature on the aboveground parts of the plants, but is, however, too much to cite in this book. |
| diagram of root structure: Parabolic Geometries I Andreas Čap, Jan Slovák, 2024-07-29 Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hypersurface type CR-structures and various types of generic distributions. The characteristic feature of parabolic geometries is an equivalent description by a Cartan geometry modeled on a generalized flag manifold (the quotient of a semisimple Lie group by a parabolic subgroup). Background on differential geometry, with a view towards Cartan connections, and on semisimple Lie algebras and their representations, which play a crucial role in the theory, is collected in two introductory chapters. The main part discusses the equivalence between Cartan connections and underlying structures, including a complete proof of Kostant's version of the Bott–Borel–Weil theorem, which is used as an important tool. For many examples, the complete description of the geometry and its basic invariants is worked out in detail. The constructions of correspondence spaces and twistor spaces and analogs of the Fefferman construction are presented both in general and in several examples. The last chapter studies Weyl structures, which provide classes of distinguished connections as well as an equivalent description of the Cartan connection in terms of data associated to the underlying geometry. Several applications are discussed throughout the text. |
| diagram of root structure: Lie Groups Beyond an Introduction Anthony W. Knapp, 2002-08-21 This book takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. The book initially shares insights that make use of actual matrices; it later relies on such structural features as properties of root systems. |
| diagram of root structure: The Structure of Affine Buildings. (AM-168) Richard M. Weiss, 2008-09-08 In The Structure of Affine Buildings, Richard Weiss gives a detailed presentation of the complete proof of the classification of Bruhat-Tits buildings first completed by Jacques Tits in 1986. The book includes numerous results about automorphisms, completions, and residues of these buildings. It also includes tables correlating the results in the locally finite case with the results of Tits's classification of absolutely simple algebraic groups defined over a local field. A companion to Weiss's The Structure of Spherical Buildings, The Structure of Affine Buildings is organized around the classification of spherical buildings and their root data as it is carried out in Tits and Weiss's Moufang Polygons. |
| diagram of root structure: Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason, 2001-06-12 A great book ... a necessary item in any mathematical library. --S. S. Chern, University of California A brilliant book: rigorous, tightly organized, and covering a vast amount of good mathematics. --Barrett O'Neill, University of California This is obviously a very valuable and well thought-out book on an important subject. --Andre Weil, Institute for Advanced Study The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been--and continues to be--the standard source for this material. Helgason begins with a concise, self-contained introduction to differential geometry. Next is a careful treatment of the foundations of the theory of Lie groups, presented in a manner that since 1962 has served as a model to a number of subsequent authors. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbb{C}$ and Cartan's classification of simple Lie algebras over $\mathbb{R}$, following a method of Victor Kac. The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All of the problems have either solutions or substantial hints, found at the back of the book. For this edition, the author has made corrections and added helpful notes and useful references. Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis. |
| diagram of root structure: Hypergeometry, Integrability and Lie Theory Erik Koelink, Stefan Kolb, Nicolai Reshetikhin, Bart Vlaar, 2022-08-30 This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7–11, 2020, which was dedicated to the 50th birthday of Jasper Stokman. The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type. |
| diagram of root structure: Biology-vol-I Dr S Venugopal, A text book on Biology |
| diagram of root structure: Groups and Manifolds Pietro Giuseppe Fré, Alexander Fedotov, 2017-12-18 Groups and Manifolds is an introductory, yet a complete self-contained course on mathematics of symmetry: group theory and differential geometry of symmetric spaces, with a variety of examples for physicists, touching briefly also on super-symmetric field theories. The core of the course is focused on the construction of simple Lie algebras, emphasizing the double interpretation of the ADE classification as applied to finite rotation groups and to simply laced simple Lie algebras. Unique features of this book are the full-fledged treatment of the exceptional Lie algebras and a rich collection of MATHEMATICA Notebooks implementing various group theoretical constructions. |
| diagram of root structure: Lie Algebras and Algebraic Groups Patrice Tauvel, Rupert W. T. Yu, 2005-04-25 Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters. |
| diagram of root structure: Matrix Groups Andrew Baker, 2012-12-06 This book offers a first taste of the theory of Lie groups, focusing mainly on matrix groups: closed subgroups of real and complex general linear groups. The first part studies examples and describes classical families of simply connected compact groups. The second section introduces the idea of a lie group and explores the associated notion of a homogeneous space using orbits of smooth actions. The emphasis throughout is on accessibility. |
| diagram of root structure: Representation Theory of Symmetric Groups Pierre-Loic Meliot, 2017-05-12 Representation Theory of Symmetric Groups is the most up-to-date abstract algebra book on the subject of symmetric groups and representation theory. Utilizing new research and results, this book can be studied from a combinatorial, algorithmic or algebraic viewpoint. This book is an excellent way of introducing today’s students to representation theory of the symmetric groups, namely classical theory. From there, the book explains how the theory can be extended to other related combinatorial algebras like the Iwahori-Hecke algebra. In a clear and concise manner, the author presents the case that most calculations on symmetric group can be performed by utilizing appropriate algebras of functions. Thus, the book explains how some Hopf algebras (symmetric functions and generalizations) can be used to encode most of the combinatorial properties of the representations of symmetric groups. Overall, the book is an innovative introduction to representation theory of symmetric groups for graduate students and researchers seeking new ways of thought. |
| diagram of root structure: Lie Groups, Lie Algebras, and Representations Brian C. Hall, 2003-08-07 This book provides an introduction to Lie groups, Lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a useful addition to the literature. First, it treats Lie groups (not just Lie alge bras) in a way that minimizes the amount of manifold theory needed. Thus, I neither assume a prior course on differentiable manifolds nor provide a con densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating the representation theory of SU(2) and SU(3) in detail before going to the general case. This allows the reader to see roots, weights, and the Weyl group in action in simple cases before confronting the general theory. The standard books on Lie theory begin immediately with the general case: a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time. |
| diagram of root structure: Algebraic Groups and their Representations R.W. Carter, J. Saxl, 2012-12-06 This volume contains 19 articles written by speakers at the Advanced Study Institute on 'Modular representations and subgroup structure of al gebraic groups and related finite groups' held at the Isaac Newton Institute, Cambridge from 23rd June to 4th July 1997. We acknowledge with gratitude the financial support given by the NATO Science Committee to enable this ASI to take place. Generous financial support was also provided by the European Union. We are also pleased to acknowledge funds given by EPSRC to the Newton Institute which were used to support the meeting. It is a pleasure to thank the Director of the Isaac Newton Institute, Professor Keith Moffatt, and the staff of the Institute for their dedicated work which did so much to further the success of the meeting. The editors wish to thank Dr. Ross Lawther and Dr. Nick Inglis most warmly for their help in the production of this volume. Dr. Lawther in particular made an invaluable contribution in preparing the volume for submission to the publishers. Finally we wish to thank the distinguished speakers at the ASI who agreed to write articles for this volume based on their lectures at the meet ing. We hope that the volume will stimulate further significant advances in the theory of algebraic groups. |
| diagram of root structure: Representations of Reductive Groups Roger W. Carter, Meinolf Geck, 1998-09-03 This volume provides a very accessible introduction to the representation theory of reductive algebraic groups. |
| diagram of root structure: Algebraic Groups: Structure and Actions Mahir Bilen Can, 2017-04-06 This volume contains the proceedings of the 2015 Clifford Lectures on Algebraic Groups: Structures and Actions, held from March 2–5, 2015, at Tulane University, New Orleans, Louisiana. This volume consists of six articles on algebraic groups, including an enhanced exposition of the classical results of Chevalley and Rosenlicht on the structure of algebraic groups; an enhanced survey of the recently developed theory of pseudo-reductive groups; and an exposition of the recently developed operational -theory for singular varieties. In addition, there are three research articles containing previously unpublished foundational results on birational automorphism groups of algebraic varieties; solution of Hermite-Joubert problem over -closed fields; and cohomological invariants and applications to classifying spaces. The old and new results presented in these articles will hopefully become cornerstones for the future development of the theory of algebraic groups and applications. Graduate students and researchers working in the fields of algebraic geometry, number theory, and representation theory will benefit from this unique and broad compilation of fundamental results on algebraic group theory. |
| diagram of root structure: Handbook of Geometry and Topology of Singularities I José Luis Cisneros Molina, Dũng Tráng Lê, José Seade, 2020-10-24 This volume consists of ten articles which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject. This is the first volume in a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook. |
| diagram of root structure: A Spectral Theory Of Noncommuting Operators Rongwei Yang, |
| diagram of root structure: D-Modules, Perverse Sheaves, and Representation Theory Kiyoshi Takeuchi, Ryoshi Hotta, Toshiyuki Tanisaki, 2007-10-12 D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory. |
| diagram of root structure: Root Systems of Certain Desert Plants ... Millard S. Markle, 1917 |
| diagram of root structure: Applying the Classification of Finite Simple Groups Stephen D. Smith, 2018-04-30 Classification of Finite Simple Groups (CFSG) is a major project involving work by hundreds of researchers. The work was largely completed by about 1983, although final publication of the “quasithin” part was delayed until 2004. Since the 1980s, CFSG has had a huge influence on work in finite group theory and in many adjacent fields of mathematics. This book attempts to survey and sample a number of such topics from the very large and increasingly active research area of applications of CFSG. The book is based on the author's lectures at the September 2015 Venice Summer School on Finite Groups. With about 50 exercises from original lectures, it can serve as a second-year graduate course for students who have had first-year graduate algebra. It may be of particular interest to students looking for a dissertation topic around group theory. It can also be useful as an introduction and basic reference; in addition, it indicates fuller citations to the appropriate literature for readers who wish to go on to more detailed sources. |
| diagram of root structure: Linear Algebraic Monoids Lex E. Renner, 2005-12-08 This solid volume discusses all the key topics in detail, including classification, orbit structure, representations, universal constructions, and abstract analogues. Open problems are discussed as they arise and many useful exercises are included. |
| diagram of root structure: Foundations of Computational Mathematics, Budapest 2011 Felipe Cucker, Teresa Krick, Allan Pinkus, Agnes Szanto, 2012-11-15 The Foundations of Computational Mathematics meetings are a platform for cross-fertilisation between numerical analysis, mathematics and computer science. This volume is a collection of articles based on plenary presentations, given at the 2011 meeting, by some of the world's foremost authorities in computational mathematics. The topics covered reflect the breadth of research within the area as well as the richness of interactions between seemingly unrelated branches of pure and applied mathematics. As a result this volume will be of interest to researchers in the field of computational mathematics and also to non-experts who wish to gain some insight into the state of the art in this active and significant field. |
| diagram of root structure: A Tour of Representation Theory Martin Lorenz, 2018 Offers an introduction to four different flavours of representation theory: representations of algebras, groups, Lie algebras, and Hopf algebras. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable the reader to pursue research in representation theory. |
| diagram of root structure: Locally Mixed Symmetric Spaces Bruce Hunt, 2021-09-04 What do the classification of algebraic surfaces, Weyl's dimension formula and maximal orders in central simple algebras have in common? All are related to a type of manifold called locally mixed symmetric spaces in this book. The presentation emphasizes geometric concepts and relations and gives each reader the roter Faden, starting from the basics and proceeding towards quite advanced topics which lie at the intersection of differential and algebraic geometry, algebra and topology. Avoiding technicalities and assuming only a working knowledge of real Lie groups, the text provides a wealth of examples of symmetric spaces. The last two chapters deal with one particular case (Kuga fiber spaces) and a generalization (elliptic surfaces), both of which require some knowledge of algebraic geometry. Of interest to topologists, differential or algebraic geometers working in areas related to arithmetic groups, the book also offers an introduction to the ideas for non-experts. |
| diagram of root structure: Simple Groups of Lie Type Roger W. Carter, 1989-01-18 Now available in paperback--the standard introduction to the theory of simple groups of Lie type. In 1955, Chevalley showed how to construct analogues of the complex simple Lie groups over arbitrary fields. The present work presents the basic results in the structure theory of Chevalley groups and their twisted analogues. Carter looks at groups of automorphisms of Lie algebras, makes good use of Weyl group (also discussing Lie groups over finite fields), and develops the theory of Chevalley and Steinberg groups in the general context of groups with a (B,N)-pair. This new edition contains a corrected proof of the simplicity of twisted groups, a completed list of sporadic simple groups in the final chapter and a few smaller amendments; otherwise, this work remains the classic piece of exposition it was when it first appeared in 1971. |
| diagram of root structure: Lectures On Lie Groups (Second Edition) Wu-yi Hsiang, 2017-04-07 This volume consists of nine lectures on selected topics of Lie group theory. We provide the readers a concise introduction as well as a comprehensive 'tour of revisiting' the remarkable achievements of S Lie, W Killing, É Cartan and H Weyl on structural and classification theory of semi-simple Lie groups, Lie algebras and their representations; and also the wonderful duet of Cartan's theory on Lie groups and symmetric spaces.With the benefit of retrospective hindsight, mainly inspired by the outstanding contribution of H Weyl in the special case of compact connected Lie groups, we develop the above theory via a route quite different from the original methods engaged by most other books.We begin our revisiting with the compact theory which is much simpler than that of the general semi-simple Lie theory; mainly due to the well fittings between the Frobenius-Schur character theory and the maximal tori theorem of É Cartan together with Weyl's reduction (cf. Lectures 1-4). It is a wonderful reality of the Lie theory that the clear-cut orbital geometry of the adjoint action of compact Lie groups on themselves (i.e. the geometry of conjugacy classes) is not only the key to understand the compact theory, but it actually already constitutes the central core of the entire semi-simple theory, as well as that of the symmetric spaces (cf. Lectures 5-9). This is the main reason that makes the succeeding generalizations to the semi-simple Lie theory, and then further to the Cartan theory on Lie groups and symmetric spaces, conceptually quite natural, and technically rather straightforward. |
| diagram of root structure: Standard Monomial Theory V. Lakshmibai, K. N. Raghavan, 2007-12-23 Schubert varieties provide an inductive tool for studying flag varieties. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties on the other. |
| diagram of root structure: Natural Biodynamics Vladimir G. Ivancevic, Tijana T. Ivancevic, 2005 This comprehensive volume is a graduate-level text in human biodynamics, written in the unified categorical language of modern differential geometry and topology. Combining mathematics, physics and robotics with human physiology, this is the first book that describes all levels of human biodynamics, from musculo-skeletal mechanics to the higher brain functions. The book develops and uses a variety of research methods, ranging from chaos theory and Haken's synergetics, through quantum mechanics, to nonlinear control and artificial intelligence, to provide the means to understand, predict and control the behavior of human-like systems in their full neuro-musculo-skeletal complexity. The applications of this unique scientific methodology range from prediction of human neuro-musculo-skeletal injuries to brain-like control of humanoid robots. |
| diagram of root structure: Diagram Geometry Francis Buekenhout, Arjeh M. Cohen, 2013-01-26 This book provides a self-contained introduction to diagram geometry. Tight connections with group theory are shown. It treats thin geometries (related to Coxeter groups) and thick buildings from a diagrammatic perspective. Projective and affine geometry are main examples. Polar geometry is motivated by polarities on diagram geometries and the complete classification of those polar geometries whose projective planes are Desarguesian is given. It differs from Tits' comprehensive treatment in that it uses Veldkamp's embeddings. The book intends to be a basic reference for those who study diagram geometry. Group theorists will find examples of the use of diagram geometry. Light on matroid theory is shed from the point of view of geometry with linear diagrams. Those interested in Coxeter groups and those interested in buildings will find brief but self-contained introductions into these topics from the diagrammatic perspective. Graph theorists will find many highly regular graphs. The text is written so graduate students will be able to follow the arguments without needing recourse to further literature. A strong point of the book is the density of examples. |
| diagram of root structure: Lie Groups Daniel Bump, 2004-06-17 This book proceeds beyond the representation theory of compact Lie groups (which is the basis of many texts) and offers a carefully chosen range of material designed to give readers the bigger picture. It explores compact Lie groups through a number of proofs and culminates in a topics section that takes the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as unifying them. |
| diagram of root structure: A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods Johan G. F. Belinfante, Bernard Kolman, 1989-01-01 In this reprint edition, the character of the book, especially its focus on classical representation theory and its computational aspects, has not been changed |
| diagram of root structure: Group Representation for Quantum Theory Masahito Hayashi, 2016-11-18 This book explains the group representation theory for quantum theory in the language of quantum theory. As is well known, group representation theory is very strong tool for quantum theory, in particular, angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, quark model, quantum optics, and quantum information processing including quantum error correction. To describe a big picture of application of representation theory to quantum theory, the book needs to contain the following six topics, permutation group, SU(2) and SU(d), Heisenberg representation, squeezing operation, Discrete Heisenberg representation, and the relation with Fourier transform from a unified viewpoint by including projective representation. Unfortunately, although there are so many good mathematical books for a part of six topics, no book contains all of these topics because they are too segmentalized. Further, some of them are written in an abstract way in mathematical style and, often, the materials are too segmented. At least, the notation is not familiar to people working with quantum theory. Others are good elementary books, but do not deal with topics related to quantum theory. In particular, such elementary books do not cover projective representation, which is more important in quantum theory. On the other hand, there are several books for physicists. However, these books are too simple and lack the detailed discussion. Hence, they are not useful for advanced study even in physics. To resolve this issue, this book starts with the basic mathematics for quantum theory. Then, it introduces the basics of group representation and discusses the case of the finite groups, the symmetric group, e.g. Next, this book discusses Lie group and Lie algebra. This part starts with the basics knowledge, and proceeds to the special groups, e.g., SU(2), SU(1,1), and SU(d). After the special groups, it explains concrete applications to physical systems, e.g., angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, and quark model. Then, it proceeds to the general theory for Lie group and Lie algebra. Using this knowledge, this book explains the Bosonic system, which has the symmetries of Heisenberg group and the squeezing symmetry by SL(2,R) and Sp(2n,R). Finally, as the discrete version, this book treats the discrete Heisenberg representation which is related to quantum error correction. To enhance readers' undersnding, this book contains 54 figures, 23 tables, and 111 exercises with solutions. |
| diagram of root structure: Lie Groups J.J. Duistermaat, Johan A.C. Kolk, 2012-12-06 This (post) graduate text gives a broad introduction to Lie groups and algebras with an emphasis on differential geometrical methods. It analyzes the structure of compact Lie groups in terms of the action of the group on itself by conjugation, culminating in the classification of the representations of compact Lie groups and their realization as sections of holomorphic line bundles over flag manifolds. Appendices provide background reviews. |
| diagram of root structure: Singularity Theory I V.I. Arnold, V.V. Goryunov, O.V. Lyashko, V.A. Vasil'ev, 1998-03-17 This is a compact guide to the principles and main applications of Singularity Theory by one of the world’s top research groups. It includes a number of new results as well as a carefully prepared and extensive bibliography that makes it easy to find the necessary details. It’s ideal for any mathematician or physicist interested in modern mathematical analysis. |
| diagram of root structure: Lie Algebraic Methods in Integrable Systems Amit K. Roy-Chowdhury, 2021-01-04 Over the last thirty years, the subject of nonlinear integrable systems has grown into a full-fledged research topic. In the last decade, Lie algebraic methods have grown in importance to various fields of theoretical research and worked to establish close relations between apparently unrelated systems. The various ideas associated with Lie algebra and Lie groups can be used to form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author exposes the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as an effective tool. The book begins by establishing a practical basis in Lie algebra, including discussions of structure Lie, loop, and Virasor groups, quantum tori and Kac-Moody algebras, and gradation. It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approach-for both finite and infinite dimension Lie algebra. The author also presents the modern approach to symmetries of integrable systems, including important new ideas in symmetry analysis, such as gauge transformations, and the soldering approach. He then moves to Hamiltonian structure, where he presents the Drinfeld-Sokolov approach, the Lie algebraic approach, Kupershmidt's approach, Hamiltonian reductions and the Gelfand Dikii formula. He concludes his treatment of Lie algebraic methods with a discussion of the classical r-matrix, its use, and its relations to double Lie algebra and the KP equation. |
| diagram of root structure: Canadian Journal of Mathematics , 1994 |
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